Dispersive non-Hermitian optical heterostructures
1. INTRODUCTION
The discovery of parity-time symmetric (PT-symmetric) optics has prompted a surge of research activities for engineering synthetic materials with new properties and functionalities. PT-symmetric systems are non-Hermitian, but can exhibit entirely real spectra as long as they respect the conditions of PT symmetry [1]. PT-symmetric photonic systems [2–
It is necessary to note that in practice it is difficult to design structures with perfectly balanced gain and loss. The aim of this paper is to explore the fundamental properties of one-dimensional (1D) layered non-Hermitian systems (with or without PT-symmetry) by taking into account the dispersion for gain and loss media. We derive the characteristic frequencies for practical realization of PT-symmetry, and we investigate the role of material dispersion in non-Hermitian scattering systems, where the spatial distributions of gain and loss are not subject to any spatial symmetry requirements. We point out the existence of exceptional points (EPs) associated with degenerate scattering eigenstates.
2. THEORETICAL ANALYSIS
Let us consider the bilayer composed of two slabs of thicknesses,
The gain and loss media are described by the Lorentzian permittivity. The dielectric permittivity components for the layer of type
First, let us consider a PT-symmetric optical system. In this case, we assume that two slabs have an identical thickness,
We use positive parameters
The dependence of the absorption coefficient on the frequency is presented in Fig.
Fig. 2. Absorption coefficient and parameter as the functions of frequency at (solid red line), (dashed black line), and (dash-dot blue line).
Substitution of Eq. (
From Eq. (
Since the layers are assumed to be homogeneous in the
Wave scattering in the proposed system is modeled using the corresponding optical scattering matrix (
The eigenvalues of the
3. NUMERICAL RESULTS
The reflectance for both left and right incidence and the transmittance of TM waves incident at angle
Fig. 3. (a) Reflectance [ - solid red line and - dashed blue line] and transmittance (solid black line) of TM wave incident at on 1D system with , , , , , , , , and . (b) Real (solid lines) and imaginary (dashed lines) parts of dielectric permittivities of the layers near emission frequencies of the gain (red lines) and loss (black lines) regions. (c) Modulus of eigenvalues of -matrix for 1D system as a function of frequency; red and blue lines correspond to the different eigenvalues of the same scattering matrix. The range of frequencies in (a) and (c) corresponding to the loss dominated system is gray shaded.
Inspired by the so-called, “loss-induced transparency” phenomena [4], we examine now the increase of the transmittance in a loss-dominated system [a system for which
Fig. 4. (a) Reflectance [ - solid red line and - dashed blue line] and transmittance (solid black line) of TM wave incident at and on non-Hermitian system with , , , , , , , and as a function of absorption coefficient. (b) Distribution of the energy flux density of an electromagnetic field in the system with the default parameters and (solid green line) and (dashed brown line); the area corresponding to gain material is red shaded, the area corresponding to loss material is green shaded. (c), (d) Transmittance of TM wave incident at (solid black line), (dash-dot green line), (dashed brown line), and . The range of frequencies in (a), (c), and (d) corresponding to loss dominated system is gray shaded.
To further investigate this effect of amplification in a loss-dominated multilayer heterostructure, the transmittance coefficient
Fig. 5. (a) Geometry of non-Hermitian periodic stack. (b), (c) Geometries of non-Hermitian random stacks. (d), (e) Transmittance of TM wave through the stacks incident at and (d) , (e) ; dash-dot black curve, four layers with , geometry of the system is presented in (a); dashed blue curves, six layers with thicknesses and , geometry of the stack is presented in (b); solid green curves, six layers with thicknesses , , and , geometry of the stack is presented in (c). The relevant layer parameters for the stacks are the same as in Fig. 4 . The range of frequencies in (d) and (e) corresponding to loss dominated system is gray shaded.
As mentioned before, in PT-symmetric systems the eigenvalues of the scattering matrix identify the exceptional points at which the symmetry-breaking transitions occur. To illustrate the effect of the gain/absorption coefficient, the dependence of
Fig. 6. Eigenvalues of the -matrix as a function of absorption coefficient for (a) PT-symmetric bilayer with , , , and at (solid and dashed lines of the same color correspond to the real and imaginary parts of eigenvalues) and (b) non-Hermitian bilayer with , , , , , , , and at and . Red and blue lines correspond to the eigenvalues and . Vertical dashed lines indicate the position of EPs.
In the general case of a non-Hermitian system, the analysis of the eigenvalue spectrum dependence on the parameter
4. CONCLUSIONS
In summary, the basic scattering properties of dispersive non-Hermitian systems, where the spatial distributions of gain and loss are not subject to any spatial symmetry requirements, are systematically examined. We demonstrate that the proper combination of the parameters of constitutive materials of the system helps to implement the PT symmetry at desirable frequencies of incident waves. It is shown that the dispersive system with nonidentical parameters of materials with gain and loss could be PT-symmetric maximum for two real frequencies. One of our main results is that, for a frequency range close to the emission frequency of the gain layer by changing the parameters of the layers and incident waves and composition of the stack, we can have amplification of a transmitted wave in an on-average lossy non-Hermitian structure. The analysis of the eigenvalue spectrum of non-PT symmetric systems demonstrates the existence of EPs associated with degenerate scattering eigenstates.
O. V. Shramkova, K. G. Makris, D. N. Christodoulides, G. P. Tsironis. Dispersive non-Hermitian optical heterostructures[J]. Photonics Research, 2018, 6(4): 040000A1.